Uniform syndeticity in multiple recurrence

The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers $d,l\geq 1$ and any $\varepsilon> 0$ , we prove the existence of $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that the following holds: Consider a solvable group $\Gamma $ of derived length l, a probability space $(X, \mu )$ , and d pairwise commuting measure-preserving $\Gamma $ -actions $T_1, \ldots , T_d$ on $(X, \mu )$ . Let E be a measurable set in X with $\mu (E) \geq \varepsilon $ . Then, K many (left) translates of $$ \begin{align*} \big\{\gamma\in\Gamma\colon \mu(T_1^{\gamma^{-1}}(E)\cap T_2^{\gamma^{-1}} \circ T^{\gamma^{-1}}_1(E)\cap \cdots \cap T^{\gamma^{-1}}_d\circ T^{\gamma^{-1}}_{d-1}\circ \cdots \circ T^{\gamma^{-1}}_1(E))\geq \delta \big\} \end{align*} $$ cover $\Gamma $ . This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers $d,l\geq 1$ and any $\varepsilon> 0$ , there are $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that for all finite solvable groups G of derived length l and any subset $E\subset G^d$ with $m^{\otimes d}(E)\geq \varepsilon $ (where m is the uniform measure on G), we have that K-many (left) translates of $$ \begin{align*} \{g\in G\colon &m^{\otimes d}(\{(a_1,\ldots,a_n)\in G^d\colon \\ & (a_1,\ldots,a_n),(ga_1,a_2,\ldots,a_n),\ldots,(ga_1,ga_2,\ldots, ga_n)\in E\})\geq \delta \} \end{align*} $$ cover G. The proof of our main result is a consequence of an ultralimit version of Austin’s amenable ergodic Szeméredi theorem.